Question

a. i. If a game has a dominant strategy equilibrium, does it have a Nash equilibrium?

ii. If a game has a Nash equilibrium, does it have a dominant strategy equilibrium?

iii. If one firm has a dominant strategy, can another firm take advantage of that fact in deciding on its optimal strategy?

iv. Can a game have more than one dominant strategy equilibrium?

v. Can a game have more than one Nash equilibrium?

b) There are only two firms in the widget industry.

The total demand for widgets is Q 5 30 ? 2P.

The two firms have identical cost functions, TC 5 3 + 10Q.

The two firms agree to collude and act as though the industry were a monopoly.

At what price and quantity will this cartel maximize its profit?

Answer 1

(a) (i) If a game has a dominant strategy equilibrium , then it must have a Nash equilibrium because nash equilibriumis the strategy profiles that are mutual best reponses and if players have dominant strategy equilibrium then this implies that both players best reponse are known and which indicates the Nash equilibrium too.

(ii) If a game has a Nash equilibrium , then it does not necessarily have a dominant strategy equilibrium because it can be the case where one player has the dominant strategy and other player don't has dominant strategy but still the game has a Nash equilibrium because of their match of best reponse given what other choose.

(iii) Yes because if one player has dominant strategy then other player can choose his strategy accordingly.

(iv) No , A game can have only one dominant strategy equilibrium.

(v) Yes, a game can have more tha one Nash equilibrium.